Rashi seems to suggest that is was vertical:
זקיפא - פורק"אבלעזבצדאחדזהלמטהמזה
זקיפא - Forche in [old French]. On one side [of it], one below the other.
Exploring this a little bit further, this topic makes an appearance in Tosafos s.v. "b'rum shalosh amos" on Chagigah 11a which discusses that a person is 3 amos in height, not including their head.
It writes there:
וכןמוכחבעובדאדבניהמןבתרגוםשלמגילתאסתרשנסדרעלפיהמדרששגובהקומתןג'אמותלבדהראשדקחשיבואזילעלרוםחמשיםאמותלפיתלייתכלהעשרהוהיונהרגיםכברקודםהתלייהוכןמשתעיקראבשושןהבירההרגוהיהודיםואבדחמשמאותאישואתעשרתבניהמןמשמעשבכללהריגההיווגםהפייטיסדבסליחהאדםבקוםחישבשלשאמותוהרביעית (וחמישית) אוירמגולה
This is clear from the episode with Bnei Haman in the Targum of Megilas Esther; it is arranged according to the Midrash that their height was three Amos without the head, for it calculates up to the height of 50 Amos based on the hanging of all 10, and they were already killed before the hanging; The verses say so. "In Shushan the capital alone, the Jews have killed a total of five hundred men, as well as the ten sons of Haman (Esther 9:12) This connotes that they were among those killed. Also the Payat, in the Selichah "Adam b'Kum" (that Ashkenazim say on Ta'anis Esther, wrote) "[each] man occupied three Amos, and the fourth Amah was exposed air."
The Targum helps to explain it even clearer:
המןאיצטלבעלתלתאאמיןפרשנדתארחיקמיניהאמתאחדא. פרשנדתאאיצטלבעלתלתאמיןדלפוןרחיקמיניהאמתאחדא. דלפוןאיצטלבעלתלתאמיןאספתארחיקמיניהאמתאחדא. אספתאאיצטלבעלתלתאמיןפורתארחיקמיניהאמתאחדא. פורתאאיצטלבעלתלתאמיןאדליארחיקמיניהאמתאחדא. אדליאאיצטלבעלתלתאמיןארידתארחיקמיניהאמתאחדא. ארידתאאיצטלבעלתלתאמיןפרמשתארחיקמיניהאמתאחדא. פרמשתאאיצטלבעלתלתאמיןאריסירחיקמיניהאמתאחדא. אריסיאיצטלבעלתלתאמיןארידירחיקמיניהאמתאחדא. ארידיאיצטלבעלתלתאמיןויזתארחיקמיניהאמתאחדאדהואעשיראה. עשרתיבנוידהמןאיצטלבועלצליבא. והואצליבאמשקעבארעאתלתאמיןובריחהוהרםתלתאמיןמןארעאוהוהאורכיהדצליבאארבעיןוארבעהאמין. והמןצליבברישיההוהסכוםדצליבאחמשיןאמין
Haman was hanged in a space of 3 amos, with Parshandasa a distance of one amah away from him. Parshandas was hanged in a space of 3 amos, with Dalfon a distance of one amah away from him. Dalfon was hanged in a space of 3 amos, with Aspasa a distance of one amah away from him. Aspasa was hanged in a space of 3 amos, with Porasa a distance of one amah away from him. Porasa was hanged in a space of 3 amos, with Adalia a distance of one amah away from him. Adalia was hanged in a space of 3 amos, with Aridasa a distance of one amah away from him. Aridasa was hanged in a space of 3 amos, with Parmashta a distance of one amah away from him. Parmashta was hanged in a space of 3 amos, with Arisai a distance of one amah away from him. Arisai was hanged in a space of 3 amos, with Aridai a distance of one amah away from him. Aridai was hanged in a space of 3 amos, with Vayzasa a distance of one amah away from him. Vayzasa was hanged in a space of 3 amos, with an amah distance after him - which completes the 10 sons. The 10 sons were hanged on a gallow, and this gallows was submerged into the ground 3 amos, and the beam had a clearance of 3 (additional) amos from the ground, meaning that the length of the (occupied) beam was 44 amos. And Haman was hanged at the top which completes the measurement of 50 amos.
Please find a somewhat crude graphical representation of the orientation as follows:
